https://file.aiquickdraw.com/mathbot/file/1727216946499mukkhcny.jpg

The given problem involves differentiating the function \( g(x) = e^{x^2 - x} \), using the properties of exponential functions and basic differentiation rules.

### Step-by-step solution:

#### 1. Function definition:
\[ g(x) = e^{x^2 - x} \]

#### 2. Differentiation:
We need to apply the chain rule to differentiate this function. The chain rule states that:
\[ \frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x) \]
where \( u(x) = x^2 - x \).

- First, differentiate \( u(x) = x^2 - x \) with respect to \( x \):
  \[ u'(x) = 2x - 1 \]

- Now, apply the chain rule to \( g(x) \):
  \[
  g'(x) = e^{x^2 - x} \cdot (2x - 1)
  \]

#### Final expression:
Thus, the derivative of \( g(x) \) is:
\[ g'(x) = (2x - 1) \cdot e^{x^2 - x} \]

### Correct alternative:
Looking at the provided alternatives, option **b)** matches this derivative:
\[ g'(x) = (2x - 1) \cdot e^{x^2 - x} \]

### Would you like further details or have any questions?  
Here are 5 related questions to deepen understanding:

1. How does the chain rule apply to other exponential functions like \( e^{\sin(x)} \)?
2. What is the difference between differentiating polynomial and exponential functions?
3. Can you find the second derivative of \( g(x) \)?
4. How would the process differ if the function were \( g(x) = \ln(x^2 - x) \)?
5. How does the derivative of exponential functions differ from trigonometric functions?

#### Tip:
When applying the chain rule, always first identify the inner function and its derivative before multiplying by the derivative of the outer function.

Uma função exponencial da forma f(x) = e^x tem a propriedade de ser sua própria derivada, ou seja, f'(x) = e^x. Mediante essa informação, e os conhecimentos sobre as regras de derivação, considere a função: g(x) = e^(x^2 - x). Assinale a alternativa que forneça a derivada de 1ª ordem da função g(x).

Learn how to find the first derivative of the exponential function g(x) = e^(x^2 - x) using the chain rule. This step-by-step solution covers key calculus concepts, including the differentiation of exponential functions. Suitable for advanced high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation